Adaptive Kernel Estimation of the Spectral Density with Boundary Kernel Analysis

Adapti v e Kernel Estimation of th e Sp ectral Density with Boundary Ke rnel Analysis Alexander Sidorenk o and Kurt S. Riedel Abstract. A h ybrid estimator of the log-sp ectral densit y of a sta- tionary time series is prop osed. First, a m ultipl e tap er estimate is p erformed, follow ed b y kernel smo othing the log- m ultitap er estimate. This pro cedure reduces the exp ected mean square error by ( π 2 4 ) . 8 o v er simply smo othing the log tap ered p erio dogram. The optimal n um b er of tap ers is O ( N 8 / 15 ). A data adaptive implementation of a v ariable bandwidth kernel smo other is given. When the sp ectral density is di s- con tinu ous, one sided smo othing estimates are used. § 1. In tro d uction W e consider a discrete, stationary , Gaussian, time series { x j , j = 1 , . . . N } with a piecewise smo ot h sp ectral densit y , S ( f ), that is b o unded aw a y from zero. The auto co v ariance is t he F ourier t r a nsform of the sp ectral densit y: Co v [ x j , x k ] = R 1 / 2 − 1 / 2 S ( f ) e 2 π i ( j − k ) f d f . Our goal is to estimate the log sp ec- tral densit y: θ ( f ) ≡ ln[ S ( f )]. W e consider data adaptiv e k ernel smo other estimators which self-consisten tly estimate the b est lo cal halfwidth for smo othing. When the spectral density is discontin uous at a po i n t, t he k ernel estimat e must be one-sided. W e presen t new “b oundary” k ernels for one-sided estimation. § 2. V ariance of multitap e ring and kernel smo othing The m ul t iple t ap er estimate of the sp ectral density i s a quadratic es- timate of the form: ˆ S M T ( f ) = N X m,n =1 Q mn x m x n e 2 i ( m − n ) f = K X k =1 µ k      N X n =1 ν ( k ) n x n e − 2 π inf      2 , (1) where the µ k and ν ( k ) are K eigenv alues and orthonormal eigen ve ctors of Q . In the large N l imit, the sin usoidal tap ers are optimal: ν ( k ) m = q 2 N +1 sin  π k m N +1  [4]. Let µ k ≡ 1 /K . F o r these tap ers, the sp ectral esti- mate (1) reduces to Approximation Theory VII I 0 Charles K. Chu i and Larry L. Sc humak er (eds.), pp. 0—1. Copy right o c 1995 b y W o rld Scientific Publish ing Co., Inc. All rights of reproductio n in an y form reserved. ISBN 0-12-xxxxxx-x Boundary Kernel Sp e ctr al Esti mation 1 ˆ S M T ( f ) = ∆ K K X k =1 | y ( f + k ∆) − y ( f − k ∆) | 2 , (2) where y ( f ) is t he FT of { x } : y ( f ) = P N m =1 x m e − 2 π imf and ∆ ≡ 1 / (2 N +2). The sin usoidal m ultita p er estima te reduces the bi a s since the sidelob es of y ( f + k ∆) are partially cancelled by those of y ( f − k ∆). T o leading order in K / N , t he lo cal white noise appro ximati on holds: ˆ S M T ( f ) has a χ 2 2 K distribution. Note E  ln  χ 2 2 K  = ψ ( K ) − ln( K ), V ar  ln  χ 2 2 K  = ψ ′ ( K ), where ψ is the digamma function. W e no w consider kernel estimator s of t he q th deriv ative, ∂ q f S ( f ): d ∂ q f S ( f ) ≡ 1 h q +1 Z 1 / 2 − 1 / 2 κ  f ′ − f h  ˆ S M T ( f ′ ) d f ′ , (3) where κ ( f ) i s a smo ot h kerne l with support in [ − 1 , 1], and κ ( ± 1) = 0 and h is the bandwidth parameter. W e say a ke rnel i s of order ( q , p ) i f R f m κ ( f ) d f = q ! δ m,q , m = 0 , . . . , p − 1. W e denote the p t h momen t of a k ernel of order ( q , p ) b y B q ,p : R f p κ ( f ) d f = p ! B q ,p . F or function estimation ( q = 0), we use p = 2 and p = 4. T o estimate the second deriv ative, we use a k ernel of order (2,4) . The estimator (3) is a quadratic estimator as in (1) with Q mn = ˆ κ m − n P K k =1 ν ( k ) m ν ( k ) n , where ˆ κ m ≡ h − ( q +1) R κ ( f ′ h ) e imf ′ d f ′ is the FT of the k ernel. W e ev aluate t he v ariance of (3), using the lo cal white noise appro x- imation: V ar h d ∂ q f S ( f ) i /S ( f ) 2 = tr[ QQ ] = K X k,k ′ =1 N X n =1 N − n X m =1 − n ˜ κ 2 m ν ( k ) n + m ν ( k ) n ν ( k ′ ) n + m ν ( k ′ ) n (4) F or mh ≫ 1 , note that ˆ κ m ∼ O ( k ˆ κ k / ( mh ) 2 ). Since the k -th tap er has a scale length of v aria t ion of N /k , ν ( k ) n + m ≃ ν ( k ) n [1 + O ( km N )]. W e expand (4) in mk / N for | mh | < O (( N h/K ) 1 / 5 ) and drop all terms wit h | mh | > O (( N h/K ) 1 / 4 ): V ar h d ∂ q f S ( f ) i ∼ S ( f ) 2 K X k,k ′ =1 N X n =1 | ν ( k ) n | 2 | ν ( k ′ ) n | 2 ! N X m =1 ˜ κ 2 m ! . (5) The last line is v alid to O (1 / ( mh ) 4 ) + O ( K m/ N ), yi elding a final accuracy of (5) of O (( K / N h ) 4 / 5 ). F or the sinu soidal tap ers, (5) reduces to V ar h d ∂ q f S ( f ) i ∼ k κ 2 k S ( f ) 2 N h ( q +1)     1 + 1 2 K     + O ( ( K N h ) 4 / 5 ) , where k κ 2 k is the square in tegral of κ . T o calculate the lo cal bias, we T a y lor expand the sp ectral densit y . Note R 1 / 2 − 1 / 2 | f ′ | 2 | V ( k ) ( f ′ ) | 2 d f ′ = k 2 / (4 N 2 ). where V ( k ) is the FT of the sin usoidal ν ( k ) . 2 A. Sidor enko and K. S. Rie del § 3. Smo othe d l og-m u ltitap er estimat e W e define the m ultitap er estimate of the logarithm of the sp ectra l densit y b y ˆ θ M T ( f ) ≡ ln[ ˆ S M T ( f )] − [ ψ ( K ) − ln( K )] /K . B y av eragi ng the K estimates prior to taki ng the loga ri thm, we reduce b o t h the bias a nd the v ariance. The v ariance reduction factor from using ln[ ¯ ˆ S ( f )], instead of ln[ ˆ S ( f )] is K ψ ′ ( K ) /ψ ′ (1). F or larg e K , K ψ ′ ( K ) ≃ 1 + 1 2 K , so the v ariance reduction factor is asymptoti cally 6 / π 2 . W e define the smo ot hed m ultit ap er l n-spectra l estima te: d ∂ q f θ κ ( f ) b y k ernel smo othing ˆ θ M T ( f ) analog ous to Eq . (3). T o ev aluate the error in d ∂ q f θ κ ( f ), w e expand ˆ θ M T ( f ′ ) ab out θ ( f ): ˆ θ M T ( f ′ ) ≃ ln[ θ ( f )] +[ ˆ S M T ( f ) − S ( f ′ )] /S ( f ′ ), that is v ali d when K ≪ N , h ≪ 1, and N h ≫ 1. The leading order bias is Bias [ d ∂ q f θ ( f )] ≃ B q ,p ∂ p f θ ( f ) h p − q + ∂ q f [ θ ′′ + | θ ′ | 2 ( f )] K 2 24 N 2 . (6) The first term is the bias from k ernel smo othing and the second t erm is from the sinus oidal multitap er estimate. T o l eading order in 1 /K , the v ariance of d ∂ q f θ ( f ) reduces t o the same calculation as the v ari ance of d ∂ q f S ( f ). T o this order, the v ari ance inflation factor from the long t a i l of the ln[ χ 2 2 K ] distribution is not v isible. Since C o v [ ˆ θ M T ( f ′ ) , ˆ θ M T ( f ′′ )] ≤ [ ψ ′ ( K ) /K ] × Co v [ ˆ S M T ( f ′ ) , ˆ S M T ( f ′′ )], w e ha ve V ar h d ∂ q f θ ( f ) i ≈ k κ k 2 N h 2 q +1     1 + 1 2 K     2 . (7) Using (6) and (7), the exp ected asymptoti c square error (EASE) i n d ∂ q f θ : E     d ∂ q f θ ( f j ) − ∂ q f θ ( f j )    2  ≈ V ar h d ∂ q f θ ( f ) i + Bias 2 [ d ∂ q f θ ( f )] , (8) where corrections are O ( h 2( p − q )+1 ) + O ( 1 K ) + O (( K N h ) 4 / 5 ) + O ( 1 N h ). The b enefit of m ultitap eri ng (in terms of the v ariance reduction) tends rapidly to zero. Minimi zing (8) with respect to h and K yields K opt << N h opt and that to leading order h o ( f ) = " 2 q + 1 2( p − q ) k κ k 2 B 2 q ,p N | ∂ p f θ ( f j ) | 2     1 + 1 2 K     2 # 1 2 p +1 , (9) and K opt ∼ N (3 p + q +2) / (6 p +3) . F or ke rnels of order (0 , 2), this reduces to h opt ∼ N − 1 / 5 and K opt ∼ N 8 / 15 . Th us the ordering 1 ≪ K ≪ N h is justified. Th e EASE dep ends only we akl y o n K for 1 ≪ K ≪ N h while Boundary Kernel Sp e ctr al Esti mation 3 the dependence on the c hoice of bandwidth is strong. When the bandwidth, h o , satisfies (9), the leading order EASE reduces to E     d ∂ q f θ ( f j ) − ∂ q f θ ( f j )    2  ∼ | B q ,p ∂ p f θ ( f j ) | (4 q +2) (2 p +1)  k κ k 2 N  2( p − q ) (2 p +1) . (10) Th us the EASE i n estimating ∂ q f θ is prop orti onal to N − 2( p − q ) (2 p +1) . The loss de- p ends on the kernel shape through B q p ( κ ) and k κ k 2 . In Sec. 5, w e optimize the kernel shap e sub ject t o momen t constrain ts. If K = 1 ( a single tap er), the v ariance t erm in (7) should b e inflated b y a factor of π 2 6 P N n =1 ν 4 n . Th us using a mo derate level of m ultitap eri ng prior to smo othing the loga rithm reduces the EASE b y a factor of [ π 2 6 P N n =1 ν 4 n ] 4 / 5 = [ π 2 4 ] 4 / 5 , where w e use P N n =1 ν 4 n = 1 . 5. § 4. Data adapt iv e estimate In practice, θ ′′ ( f ) is unknown and we use a data adaptive multiple stage k ernel estima tor w here a pi l ot estimate of the opti mal bandwidth is p er- formed prior to estimating θ ( f ). When the spectral range is large, it is often essen ti al to allow the bandwidth to v ary lo cally as a function of frequency . If computatio nal effort i s not imp o rtan t, set K = N 8 / 15 ; o therwise we set K b y the computational budget. F or nonparametric function estimation, data adaptiv e m ultiple stage sc hemes are gi ven in [1-5]. Our sc heme for sp ectral estimation has the following steps: 0) Co mpute the F ourier transform, y ( f ) on a grid of size 2 N + 2 and ˆ θ 1 ( f ) = ln[ | y ( f + ∆) − y ( f − ∆) | 2 / 2( N + 1) ] o n a grid of size N + 1. Ev aluate ˆ θ M T ( f ) on a grid of size 2 N + 2. 1) Smo oth the tap ered log-p erio dogram, ˆ θ 1 ( f ), a kernel of order (0, 4). Cho ose the glo bal halfwidth by either the Rice crit erion [1] or b y fitting the a v erage square residual as a function of the global halfwidth to a tw o parameter mo del [3] . 2) Estima t e θ ′′ ( f ) b y kern el smo othing the m ultitap er estimate ˆ θ M T with global halfwidth h 2 , 4 . Relat e the optimal ( 2 ,4) to the optimal ( 0,4) global halfwidth using the halfwidth quotien t relati on [1,5]: h 2 , 4 = H ( κ 2 , 4 , κ 0 , 4 ) h 0 , 4 . 3) Estimate θ ( f ) using the v ariable halfwidth giv en by substituti ng ˆ θ ′′ ( f ) in to the optimal halfwidth expression o f E q. (9). An estimation sc heme has a relati v e con vergence rate of N − α if E h | ˆ θ ( f | ˆ h 0 , 2 ) − θ ( f ) | 2 i ≃  1 + O ( C 2 r N − 2 α )  E h | ˆ θ ( f | h 0 , 2 ) − θ ( f ) | 2 i , where h 0 , 2 is the optimal halfwidth and ˆ h 0 , 2 is the estimated halfwidth. Our dat a -a dapti v e metho d has a con vergence rate o f N − 4 / 5 and a relati ve con v ergence rat e: N − 2 / 9 . 4 A. Sidor enko and K. S. Rie del If m ultitap eri ng were used in step 1, then θ ( f n ) would b e correla ted at neigh b oring F ourier frequency . The m ultit a p er induced auto correla t ion w ould then bias t he estimate of the h o, 4 . halfwidth. W e use a single tap er in Step 1 a nd many tap ers in Steps 2-3. T o correct for this, w e inflate the v ariance in the (0,4) k ernel estimate. The halfwidth quotien t relation relates the opti mal hal fwidth for deriv ative estimates, ˆ h 2 , 4 to that of the (0 , 4) k ernel using (9): ˆ h 2 , 4 = H ( κ 2 , 4 , κ 0 , 4 ) ˆ h 0 , 4 , where H ( κ 2 , 4 , κ 0 , 4 ) ≡ 10 B 2 0 , 4 k κ 2 , 4 k 2 B 2 2 , 4 k κ 0 , 4 k 2 ! 1 9 π 2 P n | ν (1) n | 4 6 ! 1 9 . When ˆ θ ′′ ( f ) is v anishingl y small , the optimal halfwidth b ecomes large. Th us, ˆ h 0 , 2 needs to b e regula rized. F ollowing Riedel & Sidorenko (1994), w e determine t he si ze of the regularizati on from ˆ h 0 , 4 in the previous stage. § 5. Discontin uities and b oundary k ernels When S ( f ) or its deriv ative s are discon tin uous, the k ernel estimate needs to be one sided using only t he data to the left (or righ t) of dis- con tin uity , f disc . A simi lar problem o ccurs when we wish to estimate the sp ectral densit y near the b oundary , in our case at f = 1 / 2 or f = 0 if S ′ (0) 6 = 0. In these cases, the estimation p oint, f is not in t he center of the kernel domain, that w e denote as f i ∈ [ f disc , f disc + 2 h ]. The kernel estimate is a weigh ted av erage of the ˆ θ i : d ∂ q f θ ( f ) = P i K ( f , f i ) ˆ θ i in the frequency interv al, f i ∈ [ f disc , f disc + 2 h ]. The kernel function, K ( f , f i ) m ust sti ll sat i sfy the momen t conditions: P i K ( f , f i )( f i − f ) j = q ! δ j,q , 0 ≤ j < m . Th us K ( f , f i ) is asymmetric and cannot b e simply a func- tion of f − f i . W e no w describ e results in [6] on k ernel estimati on near a b oundary or discon tinuit y . W e assume that the estimation p oint, f sati sfies f ≥ f disc . F ar from the discontin uity , we use the standard k ernel estimate (3 ) with h o ( f ) g iv en b y (9). As f approac hes f disc , the domain of the kerne l t o uc hes the dis- con tin uity when f − f disc = h o ( f ). This p oint is the start of the b oundary region around the discon tinuit y . W e call this p oin t, “the righ t touch point”, f tp as defined b y the equation: f tp − f disc = h o ( f tp ). In the boundary regio n b et we en f disc and f tp , w e use a fixed halfwidth, h and mo dify the k ernel shap e to satisfy the momen t conditions. W e define f ≡ f disc + h , ˜ f ≡ ( f − f disc ) /h . The “measuremen ts” are the log m ultitap er v alues, that are ev a l uated on a grid of size 2 N : ˆ θ i ≡ ˆ θ M T ( f i ≡ i/ 2 N ) W e standardize the gri d p oints: ˜ z i ≡ ( f i − f disc ) /h . The k ernel estimate is a w eig h ted a ve rage of the ˆ θ i in the frequency in t erv al, f i ∈ [ f disc , f disc + 2 h ]. The total hal fwi dth is 2 h . W e define orthonormal p olynomial s, P j on [ f disc , f disc + 2 h ] by 1 N h P i P k ( ˜ z i ) P j ( ˜ z i ) = g k δ kj where g k is a normali zation. W e hav e expanded the kernel function, K ( f , f i ) in the p olynomi als, P j : Boundary Kernel Sp e ctr al Esti mation 5 d ∂ q f θ ( f ) = X i K ( f , f i ) ˆ θ i = 1 N h q +1 X i X j b j ( f ) P j ( ˜ z i ) ˆ θ i . The momen t conditions b ecome P k C kj b k = δ q j for j = 0 , . . . , p − 1, where C kj ( f ) ≡ P i P k ( ˜ z i ) 1 j ! ( − ˜ f ) j . The mat rix C kj ( f ) is upp er t riangular. W e solve for b 0 , b 1 . . . b p − 1 : b j = δ j,q C q q for 0 ≤ j ≤ q ; and b j = − 1 C j j j − 1 X i = q C ij b i for q < j < p . Th us the moment conditi o ns prescrib e b 0 , b 1 . . . b p − 1 while the co ef- ficien ts b p , b p +1 , . . . are free parameters. The l eading order bia s equal s θ ( p ) ( f ) h p − q P p k = q C kp b k . The summation sto ps at k = p b ecause C kp = 0 for k > p . The EASE i s a quadratic function of b j : E AS E ( f ) = 1 N h 2 q +1 P k ≥ q g k b 2 k +  θ ( p ) ( f ) h p − q P p k = q C kp b k  2 . In the absence of b oundary conditions, EASE attains the minimum when b k = 0 for k > p . and the optimal v alue b p can b e easily found. W e now restrict to t he con tin uum limit when N h → ∞ and the p o i n ts are equispaced. In t his case, the discrete sums b ecome integrals and the P j b ecome Legendre functions. W e also require p = q + 2. W e seek a b oundary k ernel in the form: K ( f , f i ) = γ q h q +1 G  ˜ f , ˜ z i  , where γ q = 1 2 Q q k =1 (2 k + 1). The function G ( ˜ f , ˜ z ) is the normalize d b oundary kernel , and i t s domain is ˜ f ∈ [ − 1 , 0], ˜ z ∈ [ − 1 , 1]. Using the Legendre p ol y nomials, P j , we expand the normalized b oundary kernel: G ( ˜ f , ˜ z ) = P j b j ( ˜ f ) P j ( ˜ z ) . W e parameterize the kernel shap e by β = h h 0 ( f ) where h is the actual halfwidth and h 0 ( f ) is given in Eq. (9). Using β instead of θ ( p ) ( f ) is adv an- tageous b ecause w e are in terested in k ernels t hat ha ve a fixed halfwidth, h in the b oundary region: h ( f ) = h 0 ( f tp ) and β = h 0 ( f tp ) h 0 ( f ) Theorem. A mong al l b oundary kernels with supp ort [0 , 2 h ] , the kernel which minimizes the le adin g or der E ASE is K ( f , f i ) = γ q h q +1 G  ˜ f , ˜ z i  wher e G ( ˜ f , ˜ z i ) = P q ( ˜ z i ) + (2 q + 3 ) ˜ f P q +1 ( ˜ z i ) + (2 q + 3 ) ˜ f 2 − 1 2 q +3 (2 q +5) β 2 q +5 + 2 2 q +5 P q +2 ( ˜ z i ) , P q , P q +1 , P q +2 ar e the L e gendr e p olynomials, β = h/h 0 ( f ) [6]. F or h ( f ) = h 0 ( f ), the optimal b oundary k ernel simplifies to G ( ˜ f , ˜ z ) = P q ( ˜ z ) + (2 q + 3) ˜ f P q +1 ( ˜ z ) + ((2 q + 3) ˜ f 2 − 1) P q +2 ( ˜ z ) . A t the touch p oi n t, f = f tp ≡ f disc + h ( f tp ) with h = h 0 ( f tp ), the optimal b oundary kerne l is iden tical to the optimal i nterior kernel: 6 A. Sidor enko and K. S. Rie del G ( ˜ f , ˜ z ) = P q ( ˜ f − ˜ z ) − P q +2 ( ˜ f − ˜ z ) . Th us using the optimal b oundary kerne l guar ante es the c onti nuity of the esti mate if at the touc h p oint w e apply the optimal in terior ke rnel of the optimal halfwidth. A t the discon tin uity , ( f = f disc , ˜ f = − 1, h = h 0 (0)), the k ernel has a simple expres sion: G ( − 1 , ˜ z ) = P q ( ˜ z ) − ( 2 q +3) P q +1 ( ˜ z ) + ( 2 q +2) P q +2 ( ˜ z ) . The leading order EA SE at t he b oundary is exactly 4( q + 1) 2 times lar ger than for the optimal interior kernel . One metho d o f constructing ke rnel shapes is to p erform a lo cal p oly- nomial regression (LPR) i n the neigh b orho o d of f . W e mo del θ ( f ) as a p th order p ol ynomial: θ ( f i ) : = P p − 1 j =0 a j ( f i − f ) j in the region f i ∈ [ f disc , f disc + 2 h ]. T he p free parameters { a j } are c hosen by minimizi ng F ( a 0 , a 1 , . . . , a p − 1 ) = X i w i ( f )   p − 1 X j =0 a j ( f i − f ) j − θ i   2 . W e take q ! a q as the estimate of θ ( q ) ( f ). The w eigh ting functions, w i ( f ), are arbit rary and w e choose them to minimize the EASE. The equiv- alence of LPR and k ernel smo othing is given b y Theorem. A kernel of typ e ( q , p ) is the e quivalent kernel of lo c al p olyno- mial r e gr e ssion of or der p − 1 with non-ne gati ve weights if a nd only if the kernel has no mor e than p − 1 sig n change s. It is kno wn (M ¨ ull er (1987), F an(1993)) that the optimal interior k ernel of t yp e ( q , p ), p − q ≡ 0 mo d 2, i n the con tinuu m limit , is pro duced by the scaling w eigh t function W ( y ) = 1 − y 2 . This c hoice is not unique! Theorem L et p − q b e even. I f data p oi nts, f i , in the interval of supp ort, [ f − h, f + h ] , ar e symmetric ar ound the estimation p o int, t , and their weights ar e chosen as w i = W  f i − f h  , then e ach of the functions W 1 ( z ) = 1 − z , W 2 ( z ) = 1 + z , W 3 ( z ) = 1 − z 2 pr o duc es the same estimator [6] . Because of the optimality i n the interior, the Bart lett-Priestley w eigh t- ing, W ( z ) = 1 − z 2 , is used often i n the b oundary region a s w ell. This do es not minimize the EASE [6]: Theorem. The asymptotic al ly optimal ke rnel is e quivalent to a l inear , nonne gativ e wei ghting function. A t the b oundary, the e q uivalent weighting e quals 2 h − ( z − f disc ) . F or the interme diate estimation p oi nts, f disc < f < f tp , the s lop e of the weighti ng line varies as t changes. F or q = 0 , the e quivalent weighting is (1 − ˜ f 2 ) h +  ˜ f + q 1 − 3 ˜ f 2 + 3 ˜ f 4  h ˜ z . § 6. Summary W e hav e analyzed the exp ected asymptotic square error of the smoothed log multitap ered p erio dogram and sho wn that multitap ering reduces t he Boundary Kernel Sp e ctr al Esti mation 7 error b y a factor of [ π 2 4 ] . 8 for t he sinusoidal tap ers. The optimal rate o f presmo othing prior t o tak ing logari thms is K ∼ N 8 / 15 , but the ex p ected loss depends only w eakly on K when 1 ≪ K ≪ N h . W e hav e prop osed a data-adaptive multiple stage v ar i able halfwidth k ernel smoot her. It has a relati v e con v ergence of N − 2 / 9 , whic h can b e impro ved t o N − 1 / 4 if desired. The multiple stage esti mate has the following steps. 0) Estimate ˆ θ M T ( f ) ≡ ln[ ˆ S M T ( f )] − B K /K as describ ed in Sec. 2. 1) Estimate the optimal kern el halfwi dth for a k ernel of (0, 4) for the l og- single tap ered p erio dogram. 2) Esti mate θ ′′ ( f ) using a ke rnel smo other of order (2,4). 3) Estimate θ ( f ) using a ke rnel smo other of order (0,2) with the halfwidth h 0 ( f ). The halfwidth is the ev aluati on of the asymptot i cally optimal halfwidth: h ( f ) ∼ c | ∂ 2 f θ | − 2 / 5 N − 1 / 5 . Ac kno wled gmen ts. W ork funded b y U.S. Dept. of Energy Gra n t DE- F G02-86ER53223. Reference s 1. M ¨ uller, H . , and U. Stadtm ¨ uller, V ariable bandwidth kerne l estimators of regression curv es, Annals of Statistics 15 (1987) 182–20 1 . 2. Riedel, K. S., Kernel estimation of the instantaneous frequency , I .E .E.E T r ans. on Signal Pr o c essi ng 42 (1994), 2644-9. 3. Riedel, K. S., a nd A. Sidorenko, Dat a Adaptive Kernel Smo others- Ho w m uc h smoo t hing? Computers i n P hysics 8 (1994) 402–409. 4. Riedel, K. S., and A. Sidorenk o, Minim um bias m ultiple tap er sp ectral estimation, I.E. E.E. T r ans. on Signal Pr o c essing 43 ( 1 995) 188-195. 5. Riedel, K. S., a nd A. Sidorenk o, Smo othed m ulti pl e tap er sp ectral estimation with data adaptive implemetat ion, Submitted. 6. Sidorenko, A. and K. S. Riedel, Optimal b oundary kernels and w eigh t- ings, Submitted. A. Sidor enko New Y ork Univ ersity , Courant Instit ute 251 Mercer St., New Y ork, NY 10012 sidorenk@cims.nyu.edu K.S. R ie del New Y ork Univ ersity , Courant Instit ute 251 Mercer St., New Y ork, NY 10012 riedel@cims.nyu.edu